__Biomedical signal processing (BMSP)__

It involves the representation of signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing domain. DSP incorporates subfields like: audio and speech signal processing, sonar and radar signal processing, sensor array processing, spectral estimation, statistical signal processing, digital image processing, signal processing for communications, control of systems, biomedical signal processing, seismic data processing, etc.

The aim of DSP is generally to measure, filter and/or compress continuous real-world analog signals. By sampling an analog signal using an analog-to-digital converter (ADC), we convert it to a digital form, which turns the analog signal into a stream of numbers. However, often, the required output signal is another analog output signal, which requires a digital-to-analog converter (DAC). Provided the fact that this process is more complicated than analog processing and has a discrete value range, the application of computational power to digital signal processing allows for added advantages over analog processing in many applications, such as error detection and correction in transmission as well as data compression.

__Statistical Parameters__

For the calculation of the statistical parameters of any function or signal, first of all we need to acquire the signal. Here, we have collected some of the signals in the dataset for the usage. For eg. ECG 30 BMP, ECG 22BMP, emg channel 1. The Signal statistics below are shown for ECG 100 BMP.

NM, The minimum and maximum values of the function are 0.26 and -0.09 respectively.

The mean of this signal over the time interval of 0.01 second is 0.001.

The skewness of the signal is 2.182 and the kurtosis is 8.896.

**Note**: There is a slight difference between the values obtained by mathematical calculation and those obtained by LabVIEW. This is because LabVIEW calculates the statistical parameters by taking discrete points in the signal, while mathematical calculation are done using equations of waveforms generated which are continuous in nature.

__Sampling & Aliasing__

During the sampling of a data, the problem of aliasing can occur if the Nyquist-Shanon criterion does not satisfy. In other words, if the sampling frequency is not more than or equal to twice of the signal frequency, the problem of aliasing may occur.

*Aliasing*

Consider for example: The signal frequency is 15 Hz while the sampling frequency is 20 Hz. Hence the ratio of sampling to signal frequencies becomes 1.333. The problem of aliasing occurs which is shown by the Fourier transform of the original signal and that of the sampled signal, which does not overlap.

*Non-Aliasing*

Consider for example: The signal frequency is 10 Hz while the sampling frequency is 50 Hz. Hence the ratio of sampling to signal frequency equal to 5. Thus, the problem of aliasing doesn’t occur, which is shown by the overlap of Fourier transform of the original signal and that of the sampled signal.

__Power Spectrum__

For a given signal, the power spectrum generates a plot of the section of a signal's power (energy per unit time) falling within given frequency bins. The most common way of generating a power spectrum is by using a discrete Fourier transform, but other techniques available such as the maximum entropy method can also be used.

__Histogram__

Histograms are used to plot the density of data, and also for density estimation: i’e. estimating the probability density function of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the x-axis Are all the same and equal to 1, then a histogram is similar to a relative frequency plot.

__Fast Fourier Transform:__

Since the Fourier transform gives the information about the frequency component of any signal, it is used for finding out what are the frequencies present in a stationary signal i.e. the frequency component of the signal is not changing with time.

__Discrete cosine transform__

A Discrete Cosine Transform (DCT) represents a sequence of finitely many data points in terms of a sum of cosine functions oscillating at different frequencies. DCTs are important for various applications in science and engineering, from lossy compression of audio and images (where small high-frequency components can be discarded), to spectral methods for the numerical solution of partial differential equations. The use of cosine rather than sine functions is important in these applications: for compression, it turns out that cosine functions are much more efficient (as explained below, fewer are needed to approximate a typical signal), whereas for differential equations the cosines express a particular choice of boundary conditions.

__H____ilbert Transform__

In mathematics and in signal processing, the **Hilbert transform **is a that takes a function, *u*(*t*), and produces a function, *H*(*u*)(*t*), having the same domain. This transform is named after David Hilbert, who first introduced the operator to solve a special case of the Riemann–Hilbert problem for holomorphic functions. It is a basic tool in Fourier analysis, and provides a concrete means for determining the conjugate of a given function or Fourier series. Furthermore, in harmonic analysis, it is an example of a singular integral operator, and of a Fourier multiplier. The Hilbert transform is also important in the field of signal processing where it is used to derive the analytic representation of a signal *u*(*t*).

The Hilbert transform was originally defined for periodic functions, or similarly for functions on the circle, in which case it is given by convolution with the Hilbert kernel. Generally, however, the Hilbert transform refers to a convolution with the ** **Cauchy kernel, for functions defined on the real line **R**** **(the boundary of the upper half-plane). The Hilbert transform is closely associated to the Paley–Wiener theorem, another result relating holomorphic functions in the upper half-plane and Fourier transforms of functions in the real line.

The Hilbert transform of a square wave

__Wavelet Transform__

In numerical and functional analysis, a **Discrete Wavelet Transform **(DWT) is any wavelet transform for which the wavelets are discretely sampled. Similar to the other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution i’e it captures both frequency *and *location information (location in time).

__IIR Filters__

**Infinite Impulse Response (IIR) **is a property of signal processing systems. Systems with

this property are called *IIR systems *or, when working with filter systems, as *IIR filters*. IIR systems possess an impulse response function that is non-zero over an infinite length of time. This is contrary to Finite Impulse Response (FIR) filters, which have fixed-duration impulse responses. The simplest analog IIR filter is an RC filter which is made up of a single resistor (R) feeding into a node that is being shared with a single capacitor (C). This filter has an exponential impulse response characterized by an RC time constant.

IIR filters can be implemented as either analog or digital filters. In digital IIR filters, the output feedback is immediately apparent in the output defining equations. Unlikely in FIR filters, in designing IIR filters it is necessary to carefully examine the "time zero" case in which the outputs of the filter are not clearly defined.

__FIR Filters__

To implement the lowpass FIR filter, we choose Lowpass from the ‘Filter Type’ menu. The sampling frequency and lower cut off frequency are set in accordance with the Nyquist criterion.

__Median Filter__

In signal processing, it is often quite desirable to perform some kind of noise reduction on a given image or signal. The median filter is a nonlinear digital filtering technique, that is often used to remove noise. This noise reduction is a typical pre-processing step to improve the results of further later processing (for example, edge detection on an image). Median filtering is very widely used in digital image processing because under specific conditions, it preserves edges while removing noise.

Consider that an input sine wave of frequency 30 Hz and amplitude 5V is given to a median filter. The filtered output is a modified sine wave whose peak has been clipped. Thus, the median filter removes the sharp peaks of the signal changing it into a smooth signal.

__Data Compression__

Consider for example: An input sine wave of frequency 30 Hz and amplitude 5V is given to a median filter. Let the reduction factor be 25. This means at a time, 25 data points are taken and the average is taken. Again, this point is plotted and the next 25 data points are considered and the average is taken. This process goes on repeating itself to get the compressed data.

The value -1.00978 specifies the instantaneous mean of the current reduction factor size.